---
tag: A6JV1E99V4
kind: statement
role: proposition
name: Determinant of the inverse is the reciprocal of the determinant
msc: [15A15]
uses: [0EB9N0S33C, 00KDZRW4C0]
sources:
  - url: https://github.com/davidaustinm/ula/blob/b6ea263ef23703072b7007485bb7a2d89c5f399b/src/section3-4.xml#L1128
    locator: src/section3-4.xml:1128
---

## Statement

Let $A$ be a square matrix with entries in a field, for example the real numbers, and suppose that $A$ is [[invertible|0EB9N0S33C]].
Then $\det A \neq 0$, and the [[determinant|00KDZRW4C0]] of the inverse is the reciprocal of the determinant of $A$:
$$\det(A^{-1}) = \frac{1}{\det A}.$$

## Remarks

The determinant is multiplicative, and $A A^{-1} = I$ has determinant $1$.
So $\det A \cdot \det(A^{-1}) = 1$, and the claim follows.